Object Not Following Bezier Curve Properly
The problem is that you added rigid body physics to your spaceship. They conflict with the animation.
To make the ship follow the curve either make your rigid body animated or remove rigid body physics alltogether.


But "Parent->Follow Path" will only set the object's position. Your ship will not be rotated along the curve. The better solution is using a "Follow Path" constraint. To make the constraint work correctly, you will have to clear the ship's translation by selecting it and pressing [Alt+G]. Set the forward axis of the constraint to '-Y' to make the ship point into the correct direction. The rest of the settings should be OK.


You can also edit the way the ship's upward direction is pointing by selecting the curve, going into edit mode and modifying the tilt of the curve's control points until their orientation suits you. Just select a control point and press [Ctrl+T]. By moving the mouse you can rotate the direction the curve interprets as 'Up'.